On the WDVV-equation in quantum K-theory
Alexander Givental
UC Berkeley and Caltech
0. Introduction. Quantum cohomology theory can be described in
general words as intersection theory in spaces of holomorphic curves in a
given Kähler or almost Kähler manifold X. By quantum K-theory we may
similarly understand a study of complex vector bundles over the spaces of
holomorphic curves in X. In these notes, we will introduce a K-theoretic
version of the Witten-Dijkgraaf-Verlinde-Verlinde equation which expresses
the associativity constraint of the “quantum multiplication” operation on
K ∗ (X).
Intersection indices of cohomology theory,
Z
ω1 ∧ ... ∧ ωk
[ space of curves ]
obtained by evaluation on the fundamental cycle of cup-products of cohomology classes, are to be replaced in K-theory by Euler characteristics
χ( space of curves ; V1 ⊗ ... ⊗ Vk )
of tensor products of vector bundles. The hypotheses needed in the definitions of the intersection indices and Euler characteristics — that the spaces
of curves are compact and non-singular, or that the bundles are holomorphic
— are rarely satisfied. We handle this foundational problem by restricting
ourselves throughout the notes to the setting where the problem disappears.
Namely, we will deal with the so called moduli spaces Xn,d of degree d genus
0 stable maps to X with n marked points assuming that X is a homogeneous Kähler space. Under the hypothesis, the moduli spaces Xn,d (we will
review their definition and properties when needed) are known to be compact complex orbifolds (see [9, 1]). We use their fundamental cycle [Xn,d ],
1
well-defined over Q, in the definition of intersection indices, and we use sheaf
cohomology in the definition of the Euler characteristic of a holomorphic
orbi-bundle V :
X
χ(Xn,d ; V ) :=
(−1)k dim H k (Xn,d ; Γ(V )).
1. Correlators. The WDVV-equation is usually formulated in terms of
the following generating function for correlators:
F (t, Q) =
∞
XX
Qd
d
n=0
n!
(t, ..., t)n,d.
Here d ∈ H2 (X, Z) runs the Mori cone of degrees, that is homology classes
represented by fundamental cycles of rational holomorphic curves in X, and
the correlators (φ1 , ..., φn)n,d are defined using the evaluation maps at the
marked points:
ev1 ×... × evn : Xn,d → X × ... × X.
In cohomology theory, we pull-back to the moduli space Xn,d the n cohomology classes φ1, ..., φn ∈ H ∗ (X, Q) of X and define the correlator among them
by
Z
(φ1 , ..., φn)n,d :=
[Xn,d ]
ev∗1(φ1 ) ∧ ... ∧ ev∗n (φn ).
In K-theory, we pull-back n elements φ1 , ..., φn ∈ K ∗ (X) (representable under
our restriction on X by holomorphic vector bundles or their formal differences) and put
(φ1 , ..., φn)n,d := χ(Xn,d ; ev∗1(φ1 ) ⊗ ... ⊗ ev∗n (φn )).
We will treat the series F as a formal function of t ∈ H depending on formal
parameters Q = (Q1 , ..., QBetti2 (X)), where H = H ∗ (X, Q) or H = K ∗(X).
Let {φα} be a graded basis in H ∗ (X, Q), and
Z
gαβ := hφα , φβ i =
φα ∧ φβ
[X]
denote the
matrix. Let (g αβ ) = (gαβ )−1 be the inverse matrix
P intersection
αβ
(so that (φα ⊗ 1)g (1 ⊗ φβ ) is Poincare-dual to the diagonal in X × X).
2
In quantum cohomology theory, one defines the quantum cup-product • on
the tangent space TtH by
hφα • φβ , φγ i := Fαβγ (t)
(where the subscripts on the RHS mean partial derivatives in the basis {φα}).
In the above notation the associativity of the quantum cup-product is equivalent to the WDVV-identity:
P
εε′
′
ε,ε′ Fαβεg Fε γδ is totally symmetric in α, β, γ, δ.
2. Stable maps, gluing and contraction. In order to explain the
proof of the WDVV-identity we have to discuss some properties of the moduli
spaces Xn,d (see [9, 1, 4] for more details).
We consider prestable marked curves (C, z), that is compact connected
complex curves C with at most double singular points and with n marked
points z = (z1, ..., zn) which are non-singular and distinct. Two holomorphic
maps, f : (C, z) → X and f ′ : (C ′, z′) → X, are called equivalent if they are
identified by an isomorphism (C, z) → (C ′, z′) of the curves. This definition
induces the concept of automorphism of a map f : (C, z) → X, and one calls
f stable if it has no non-trivial infinitesimal automorphisms. The moduli
spaces Xn,d consist of equivalence classes of stable maps with fixed number
n of marked points, degree d and arithmetical genus 0 (it is defined as g =
dim H 1 (C, OC )).
In plain words, the space of degree d holomorphic spheres in X with n
marked points is compactified by prestable curves which are trees of CP 1 ’s
and satisfy the stability condition: each irreducible component CP 1 mapped
to a point in X must carry at least 3 marked or singular points. Under
the hypothesis that X is a homogeneous Kähler space, the moduli space
RXn,d has the structure of a compact complex orbifold of dimension dimC X +
c (TX ) + n − 3.
d 1
In the case when X is a point the moduli spaces coincide with the
Deligne-Mumford compactifications M̄0,n of moduli spaces of configurations
of marked points on CP 1. For instance, M0,4 is the set CP 1 − {0, 1, ∞} of
legitimate values of the cross-ratio of 4 marked points on CP 1 . The compactification M̄0,4 = CP 1 fills-in the fibidden values of the cross-ratio by
equivalence classes of the reducible curves CP 1 ∪ CP 1 with one double point
and two marked point on each irreducible component.
3
For n ≥ 3, there is a natural contraction map Xn,d → M̄0,n defined by
composing the map f : (C, z) → X with X → pt (so that the components of
C carrying < 3 special points become unstable) and contracting the unstable
components. Similarly, one can define the forgetting maps fti : Xn+1,d → Xn,d
by disregarding the i-th marked point and contracting the component if it
has become unstable.
In particular, we will make use of the contraction map
ct : Xn+4,d → M̄0,4
defined by forgetting the map f : (C, z) → X and all the marked points
except the first four. A legitimate value λ = ct[f] of the cross-ratio means
the following: the curve C has a component C0 = CP 1 carrying 4 special
points with the cross-ratio λ, and the first 4 marked point are situated on the
branches of the tree connected to C0 at those 4 special points. A forbidden
value ct[f] = 0,1 or ∞ means that C containing a chain C0 , ..., Ck of k > 0
of CP 1 ’s such that 2 of the 4 branches of the tree carrying the marked points
are connected to the chain via C0 , and the other two — via Ck . Such stable
maps form a stratum of codimension k in the moduli space Xn,d . We will
refer to them as strata (or stable maps) of depth k.
A stable map of depth 1 is glued from 2 stable maps obtained by disconnecting C0 from C1. This gives rise to the gluing map
Xn0 +3,d0 ×∆ Xn1 +3,d1 → Xn0 +n1 +4,d0 +d1
as follows. Consider the map from Xn0 +3,d0 × Xn1 +3,d1 to X × X defined by
evaluation at the 3-rd marked points. The source of the gluing map is the
preimage of the diagonal ∆ ⊂ X × X. 1 It consists of pairs of stable maps
which have the same image of the third marked point and which therefore
can be glued at this point into a single stable map of degree d0 + d1 with
n0 + 2 + n1 + 2 marked points.
Similarly, gluing stable maps of depth k from k + 1 stable maps subject
to k diagonal constraints at the double points of the chain C0 , ..., Ck defines
appropriate gluing maps parameterizing the strata of depth k.
3. Proof of the WDVV-identity. All points in M̄0,4 represent the
same (co)homology class. Thus the analytic fundamental cycles of the fibers
1
Note that for a homogeneous Kähler X, the evaluation map is conveniently transverse
to the diagonal in X × X.
4
ct−1 (λ) are homologous in Xn+4,d . The cohomological WDVV-identity follows from the fact that for λ = 0, 1 or ∞ the fiber ct−1 (λ) consists of strata of
depth > 0, and moreover — the corresponding gluing maps (for all splittings
d = d0 + d1 of the degree and all splittings of the n = n0 + n1 marked points),
being isomorphisms at generic points, identify the analytic fundamental cycle
of the fiber with the sum of the fundamental cycles of Xn0 +3,d1 ×∆ Xn1 +3,d2 .
This allows one to equate 3 quadratic expression of the correlators which
differ by the order of the indices α, β, γ, δ associated with the first 4 marked
points.
We leave the reader to work out some standard combinatorial details
which are needed in order to translate this argument into the WDVV-identity
for the generating function F and note only that the contraction with the
′
intersection tensor (g εε ) in the WDVV-equation takes care of the diagonal
constraint ∆ ⊂ X × X for the evaluation maps.
In K-theory, similarly, the push-forward to X × X of the structural sheaf
O∆ of the diagonal is expressed as
X
′
(φε ⊗ 1)g εε (1 ⊗ φε′ )
′
via (g ε,ε ) inverse to the “intersection matrix”
gαβ := hφα , φβ i = χ(X; φα ⊗ φβ ).
The argument justifying the WDVV-equation fails, however, since the above
gluing map to ct−1 (λ) is one-to-one only at the points of depth 1 and does
not identify the corresponding structural sheaves. Indeed, a stable map of
depth k can be glued from two stable maps in k different ways and thus
belongs to the k-fold self-intersection in the image of the gluing map.
Let us examine the variety ct−1 (λ) at a point of depth k > 1. One of
the properties of Kontsevich’s compactifications Xm,d is that after passing to
the local non-singular covers (defined by the orbifold structure of the moduli
spaces) the compactifying strata form a divisor with normal crossings [9, 1].
Moreover, analyzing (inductively in k) the local structure of the contraction
map ct : Xn+4,d → M̄0,4 near a depth-k point, one easily finds the local model
λ(x1 , ..., xk, ...) = x1 ...xk for the map ct in a suitable local coordinate system.
In this model, the components x1 = 0, ..., xk = 0 of the divisor with normal
crossings represent the strata of depth 1, their intersections xi1 = xi2 = 0 —
the strata of depth 2, etc. Denote by O the algebra of functions on our local
5
chart, so that O/(xi1 , ..., xil ), i1 < ... < il, are the algebras of functions on
the depth-l strata. We have the following exact sequence of O-modules:
0 → O/(x1 ...xk ) → ⊕O/(xi ) → ⊕O/(xi1 , xi2 ) → ⊕O/(xi1 , xi2 , xi3 ) → ....
Notice that the ⊕-terms in the sequence are the algebras of functions on the
normalized strata of depth 1, depth 2, etc. Translating this local formula
to a global K-theoretic statement about gluing maps, we conclude that in
the Grothendieck group of orbi-sheaves on Xn+4,d ), the element represented
by the structural sheaf of ct−1 (λ) for λ = 0,1 or ∞ is identified with the
structural sheaf of the corresponding alternated disjoint sum over positive
depth strata:
X
X
Xn0 +3,d0 ×∆ Xn1 +3,d1 −
Xn0 +3,d0 ×∆ Xn1 +2,d1 ×∆ Xn2 +3,d2 + ...
4. Formulation and consequences. Now we can apply the above
K-theoretic statement about the moduli spaces to our generating functions.
Introduce
1X
G(t, Q) :=
gαβ tα tβ + F (t, Q).
2
α,β
Let (Gαβ ) be the matrix inverse to (Gαβ ) = (∂α∂β G).
Theorem.
X
′
Gαβε Gεε Gεγδ is totally symmetric in α, β, γ, δ.
ε,ε′
Proof. We have rewritten
′ ′
′
Fαβεg εε Fε′ γδ − Fαβε g εµ Fµµ′ g µ ε Fε′ γδ + ...
using the famous matrix identity 1 − F + F 2 − ... = (1 + F )−1. Introduce the quantum tensor product on Tt H (with H = K ∗ (X)) by
(φα • φβ , φγ ) := Gαβγ (t),
and the metric (, ) on T H is defined by (φµ , φν ) := Gµν (t).
6
Corollary 1. The operations (, ) and • define on the tangent bundle
the structure of a formal commutative associative Frobenius algebra with the
unity 1. 2
Proof. As in the cohomology theory, it is a formal corollary of the Theorem, except that the statement about the unity 1 means that Gα,1,β = Gαβ
and follows from the simplest instance of the string equation in the K-theory:
(1, t, ..., t)n+1,d = (t, ..., t)n,d. The last equality is obvious. Indeed, the
push-forward of the constant sheaf 1 along the map ft : Xn+1,d → Xn,d
forgetting the first marked point is the constant sheaf 1 on Xn,d since the
fibers are curves C of zero arithmetic genus, g = dim H 1 (C, OC ) = 0, while
H 0 (C, OC ) = C by Liouville’s theorem. We introduce on T ∗H the 1-parametric family of connection operators
X
∇q := (1 − q)d −
(φα •)∗ dtα ∧ .
α
Corollary 2. The connections ∇q are flat for any q 6= 1.
Proof. This follows from φα •φβ = φβ •φα, d2 = 0, and ∂α (φβ •) = ∂β (φα•):
′
′′ ν
∂α (φβ •)νµ = Gµαβε Gεν − Gµβε Gεε Gε′ αε′′ Gε
is symmetric with respect to α and β due to the WDVV-identity. Proposition. The operator ∇−1 is twice the Levi-Civita connection of
the metric (Gαβ ) on T ∗H.
Proof. For a metric of the form Gαβ = ∂α ∂β G the famous explicit formulas
for the Christoffel symbols yield
2Γγαβ = [Gαε,β + Gβε,α − Gαβ,ε ]Gεγ = Gαβε Gεγ = (φβ •)γα.
Corollary 3. The metric (, ) on T H is flat.
We complete this section with a description of flat sections of the connection operator ∇q in terms of K-theoretic “gravitational descendents”. Let us
introduce the generating functions
Sαβ (t, Q) := gαβ +
X Qd
n,d
2
n!
(φα , t, ..., t,
φβ
)n+2,d ,
1 − qL
At t = 0, Q = 0 it turns into the usual multiplicative structure on K ∗ (X).
7
where the correlators are defined by
(ψ1 , ..., ψnLk )m,d := χ(Xm,d ; ev∗1 (ψ1) ⊗ ... ⊗ ev∗m (ψm ) ⊗ L⊗k ).
Here L is the line orbibundle over the moduli space Xm,d of stable maps
(C, z) → X formed by the cotangent lines to C at the last marked point
(as specified by the position of the geometrical series 1 + qL + q 2L2 + ... =
(1 − qL)−1 in the correlator).
Theorem. The matrix S := (Sµν ) is a fundamental solution to the linear
PDE system:
(1 − q)∂αS = (φα•)S .
Proof. Taking φµ , φα, φβ and φν /(1 − qL) for the content of the four
distinguished marked points in the proof of the WDVV-identity, we obtain
its generalization in the form:
′
′
Gµαε Gεε ∂β Sε′ ν = Gµβε Gεε ∂α Sε′ ν ,
or (φα •)∂β S = (φβ •)∂αS. Now it remains to put φβ = 1 and use (1−q)∂1S =
S, which is another instance of the string equation:
(1, t, ..., t, φLk)n+2,d = (t, ..., t, φ(1 + L + ... + Lk ))n+1,d .
The last relation is obtained by computing the push forward of L⊗k along
ft1 : Xn+2,d → Xn+1,d . 3
5. Some open questions.
(a) Definitions. It is natural to expect that the above results extend from
the case of homogeneous Kähler spaces X to general compact Kähler and,
even more generally, almost Kähler target manifolds.
In the Kähler case, the moduli of stable degree d genus g maps with n
marked points form compact complex orbi-spaces Xg,n,d equipped with the
3
Some details can be found in [15, 14, 11, 5]. Briefly, one identifies the fibers of ft1
with the curves underlying the stable maps f : (C, z) → X with n + 1 marked points. It is
important to realize that the pull-back L′ := ft∗1 (L) of the line bundle named L on Xn+1,d
differs from the line bundle named L on Xn+2,d . In fact, there is a holomorphic section
of Hom(L′ , L) with the divisor D defined by the last marked point zn+1 ∈ C, and the
bundle L restricted to D is trivial (while L′ |D is therefore conormal to D). Since L′ is trivial
along the fibers C, we find that H 1 (C, Lk ) = 0 and H 0 (C, Lk ) = (L′ )k ⊗H 0 (C, OC (kD)) ≃
(L′ )k (1 + (L′ )−1 + ... + (L′ )−k ).
8
intrinsic normal cone [13]. The cone gives rise [3] to an element in K-group
of Xg,n,d which should be used in the definition of K-theoretic correlators in
the same manner as the virtual fundamental cycle [Xg,n,d ] is used in quantum
cohomology theory.
The moduli space Xg,n,d can be also described as the zero locus of a section
of a bundle E → B over a non-singular space. Due to the famous “deformation to the normal cone” [3], the virtual fundamental cycle represents the
Euler class of the bundle. This description survives in the almost Kähler case
and yields a topological definition and symplectic invariance of the cohomological correlators. In K-theory, there exists a topological construction of the
push forward from B to the point based on Whitney embedding theorem and
Thom isomorphisms. We don’t know however how to adjust the construction
to our actual setting where B is non-singular only in the orbifold sense.
One (somewhat awkward) option is to define K-theoretic correlators topologically by the RHS of the Kawasaki-Riemann-Roch-Hirzebruch formula [8]
for orbi-bundles over B. This proposal deserves further study even in the
Kähler case since it may lead to a ”quantum Riemann-Roch formula”.
(b) Frobenius-like structures. Our results in Section 4 show that Ktheoretic Gromov-Witten invariants of genus 0 define on the space H =
K ∗ (X) a geometrical structure very similar to the Frobenius structure [2] of
cohomology theory, but not identical to it.
One of the lessons is that the metric tensor on H, which can be in both
cases described as Fα,1,β , is constant in cohomology theory and equal to gαβ
only by an ”accident”, but remains flat in K-theory even though it is not
constant anymore.
The translation t 7→ t + τ 1 in the direction of 1 ∈ H leaves the structure
invariant in cohomology theory, but causes multiplication by eτ in K-theory
— because of a new form of the string equation. Also, the Z-grading missing in K-theory makes an important difference. It would be interesting to
study the axiomatic structure that emerges here and to compare it with the
structure implicitly encoded by K-theory on Deligne-Mumford spaces.
(c) Deligne-Mumford spaces. When the target space X is the point, the
moduli spaces Xg,n,0 are Deligne-Mumford compactifications of the moduli
spaces of genus g Riemann surfaces with n marked points. The parallel
between cohomology and K-theory suggest several problems.
Holomorphic Euler characteristics of universal cotangent line bundles and
9
their tensor products satisfy the string and dilation equations. 4 K-theoretic
generalization of the rest of Witten – Kontsevich intersection theory [15, 10]
is unclear.
The case of genus 0 and 1 has been studied in [14, 11] and [12]. The
formula
χ(M̄0,n;
1
(1 + q1/(1 − q1) + ... + qn /(1 − qn ))n−3
)=
(1 − q1L1 )...(1 − qn Ln )
(1 − q1)...(1 − qn )
found by Y.-P. Lee [11] is analogous to the famous intersection theory result
[15, 9]
Z
1
= (x1 + ... + xn )n−3 .
[M̄0,n ] (1 − x1 c1 (L1 ))...(1 − xn c1 (Ln ))
The latter formula is a basis for fixed point computations [9, 5] in equivariant
cohomology of the moduli spaces Xn,d for toric X. As it was notices by Y.-P.
Lee, the former formula is not sufficient for similar fixed point computation
in K-theory: it requires Euler characteristics accountable for invariants with
respect to permutations of the marked points. Finding an Sn -equivariant
version of Lee’s formula is an important open problem.
(d) Computations. The quantum K-ring is unknown even for X = CP 1 .
It turns out that the WDVV-equation is not powerful enough in the absence
of grading constraints and divisor equation (see, for instance, [5]).
On the other hand, for X = CP n , it is not hard to compute the generating
functions G(t, Q) and even Sαβ (t, Q, q) at t = 0 (see [12]). In cohomology
theory, this would determine the small quantum cohomology ring due to the
divisor equation which, roughly speaking, identifies the Q-deformation at
t = 0 with the t-deformation at Q = 1 along the subspace H 2 (X, Q) ⊂ H.
No replacement for the divisor equation seems to be possible in K-theory.
At the same time, the heuristic study [6] of S 1 -equivariant geometry on
the loop space LX suggests that the generating functions S = S1,β (0, Q, q)
should satisfy certain linear q-difference equations (instead of similar linear
4
The same is true not only for X = pt (see [12]). By the way, the push forward ft∗ (L)
along ft : Xg,n+1,d → Xg,n,d , described by the dilation equation, equals H + H∗ − 2 + n.
Here H is the g-dimensional Hodge bundle with the fiber H 1 (C, OC ). This answer replaces
a similar factor 2g − 2 + n in the cohomological dilation equation, but also shows that
tensor powers of H must be included to close up the list of “observables”.
10
differential equations of quantum cohomology theory). This expectation is
supported by the example of X = CP n : Y.-P. Lee [12] finds that the generating functions are solutions to the q-difference equation Dn+1 S = QS (where
(DS)(Q) := S(Q) − S(qQ)).
In the case of the flag manifold X the generating functions S have been
identified with the so called Whittaker functions — common eigen-functions
of commuting operators of the q-difference Toda system. This result and
its conjectural generalization [7] to the flag manifolds X = G/B of complex simple Lie algebras links quantum K-theory to representation theory
and quantum groups. Originally this conjecture served as a motivation for
developing the basics of quantum K-theory.
The author is thankful to Yuan-Pin Lee and Rahul Pandharipande for
useful discussions, and to the National Science Foundation for supporting
this research.
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